The relation between discrete and continuous models

I describe some results and methods that have been developed in the last years for the study of the overall behavior of variational problems for large lattice systems (e.g. atomistic system with pair potentials, variational models of vision, resistor networks, etc.) Upon scaling, some of the questions may be equivalently stated in terms of the limit of variational problems for lattice systems as the lattice spacing tends to zero, that are described by a continuous energy. This limit must be understood in a variational sense (Gamma-limit) and is in general (quite) different from the pointwise limit.

Some issues that have been addressed are:

• determination of the scaling properties of the discrete energies that correspond to continuous energies of specific type ('elastic', 'brittle fracture', 'softening' type, etc.)
• multi-scale analysis describing 'phase-transition' and 'boundary-layer' type effects
• homogenization formulas for the effective energies, and interpretation of such formulas in terms of the Cauchy-Born hypothesis
• optimal bounds for discrete composites and comparison with continuous bounds
• percolation analysis of random discrete problems.